Fractals and the Mandelbrot Set Matt Ziemke October, 2012 Matt Ziemke Fractals and the Mandelbrot Set Outline 1. Fractals 2. Julia Fractals 3. The Mandelbrot Set 4. Properties of the Mandelbrot Set 5. Open Questions Matt Ziemke Fractals and the Mandelbrot Set What is a Fractal? "My personal feeling is that the definition of a 'fractal' should be regarded in the same way as the biologist regards the definition of 'life'." - Kenneth Falconer Common Properties 1.) Detail on an arbitrarily small scale. 2.) Too irregular to be described using traditional geometrical language. 3.) In most cases, defined in a very simple way. 4.) Often exibits some form of self-similarity. Matt Ziemke Fractals and the Mandelbrot Set The Koch Curve- 10 Iterations Matt Ziemke Fractals and the Mandelbrot Set 5-Iterations Matt Ziemke Fractals and the Mandelbrot Set The Minkowski Fractal- 5 Iterations Matt Ziemke Fractals and the Mandelbrot Set 5 Iterations Matt Ziemke Fractals and the Mandelbrot Set 5 Iterations Matt Ziemke Fractals and the Mandelbrot Set 8 Iterations Matt Ziemke Fractals and the Mandelbrot Set Heighway's Dragon Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.1 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.2 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.3 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal 1.4 Matt Ziemke Fractals and the Mandelbrot Set Matt Ziemke Fractals and the Mandelbrot Set Matt Ziemke Fractals and the Mandelbrot Set Julia Fractals 2 Step 1: Let fc : C ! C where f (z) = z + c. 1 Step 2: For each w 2 C, recursively define the sequence fwngn=0 1 where w0 = w and wn = f (wn−1): The sequence wnn=0 is referred to as the orbit of w. Step 3: "Collect" all the w 2 C whose orbit is bounded, i.e., let Kc = fw 2 C : sup jwnj ≤ M; for some M > 0g n2N and let Jc = δ(Kc ) where δ(K) is the boundary of K. Jc is called a Julia set. Matt Ziemke Fractals and the Mandelbrot Set Julia Fractals - Example Let c = 0:375 + i(0:335). Consider w = 0:1i. Then, w1 = f (w0) = f (0:1i) = (0:1i) = 0:365 + 0:335i w2 = f (w1) = f (0:365 + 0:335i) = 0:396 + 0:5796i w20 ≈ 0:014 + 0:026i 1 In fact, fwngn=0 does not converge but it is bounded by 2. So 0:1i 2 Kc . Consider x = 1. Then, x1 ≈ 1:375 + 0:335i x2 ≈ 2:153 + 1:256i x3 ≈ 3:434 + 5:745i x4 ≈ −20:843 + 39:794i x5 ≈ −1148:782 − 1658:450i So looks as though 1 2= Kc . Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 1 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 2 Matt Ziemke Fractals and the Mandelbrot Set Julia Fractal - Example, Image 3 Why the colors? Matt Ziemke Fractals and the Mandelbrot Set c=-1.145+0.25i Matt Ziemke Fractals and the Mandelbrot Set c=-0.110339+0.887262i Matt Ziemke Fractals and the Mandelbrot Set c=0.06+0.72i Matt Ziemke Fractals and the Mandelbrot Set c=-0.022803-0.672621i Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set Theorem of Julia and Fatou (1920) Every Julia set is either connected or totally disconnected. Brolin's Theorem Jc is connected if and only if the orbit of zero is bounded, i.e., if and only if 0 2 Kc . Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. A natural question to ask is... What does (n) 1 M = fc 2 C : Jc is connected g = fc 2 C : ffc (0)gn=0 is boundedg look like? Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set The Mandelbrot Set cont. Matt Ziemke Fractals and the Mandelbrot Set M is a "catalog" for the connected Julia sets. Matt Ziemke Fractals and the Mandelbrot Set Interesting Facts about M 1.)If Jc is totally disconnected then Jc is homeomorphic to the Cantor set. 2.) fc : Jc ! Jc is chaotic. 3.) Julia fractals given by c-values in a given "bulb" of M are homeomorphic. 4.) M is compact. 5.) The Hausdorff dimension of δ(M) is two. Matt Ziemke Fractals and the Mandelbrot Set Open questions about M 1.) What's the area of M? (n) 1 2.) Are there any points c 2 M so that ffc (0)gn=1 is not attracted to a cycle? 3.) Is µ(δ(M)) > 0? Where µ is the Lebesgue measure. Matt Ziemke Fractals and the Mandelbrot Set.